Question

question 6 of 10
suppose a normal distribution has a mean of 266 and a standard deviation of 12. what is ( p(x geq 278) )?

a. 0.16
b. 0.475
c. 0.84
d. 0.975

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 278$, $\mu=266$, and $\sigma = 12$.
Substitute the values into the formula: $z=\frac{278 - 266}{12}=\frac{12}{12}=1$.

Step2: Use the properties of the normal distribution

For a normal distribution, we know that the total area under the curve is 1. The empirical rule (or 68 - 95 - 99.7 rule) tells us that approximately 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. Also, the normal distribution is symmetric about the mean.
The area to the left of $z = 1$ can be found using the empirical rule or z - tables. The area to the left of $z=1$ is approximately 0.84 (since about 68% of the data is between $z=- 1$ and $z = 1$, so the area to the left of $z = 1$ is $0.5+\frac{0.68}{2}=0.84$).
To find $P(x\geq278)$, we need the area to the right of $x = 278$ (or $z = 1$). Since the total area is 1, $P(x\geq278)=1 - P(x\lt278)=1 - 0.84 = 0.16$.

Answer:

A. 0.16